Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

Singular perturbations in a class of problems of optimal control with integral convex criterion

The problem of optimal control is investigated with a linear law of motion and convex quality criterion. A small positive parameter appears in front of the derivatives of some of the unknowns in the law of motion. The behaviour of the optimal solution is studied when the small parameter approaches zero with some assumptions that are different from thos encountered in the literature.     

Semigroups of polygons whose vertices define a centered partition of ℝ n

A partition $\mathfrak{F}$ of a Euclidean space into finite subsets has subgroup property SP if the family of the convex hulls of the leaves of $\mathfrak{F}$ constitutes a subgroup with respect to the Minkowski addition. If $\mathfrak{F}$ consists of orbits of a finite linear groups then SP is equivalent to the fact that the group is a Coxeter group. In this article, this assertion is proved only under the assumption of continuity and centrality of $\mathfrak{F}$ (this means that every leaf is inscribed in some sphere centered at zero). An example is given of a noncentered partition satisfying SP (such partitions cannot be Coxeter partitions).     

Metrized Semigroups

The notion of metrized order (antimetric) on a topological group is characterized by three equivalent systems of axioms and connected with pointed locally generated semigroups. In the present paper, these notions are discussed and new results are announced. The main result is an analog of the following fact in metric geometry: every left-invariant inner metric on a Lie group is Finsler (maybe, nonholonomic). In the situation considered, a norm is replaced by an antinorm, and a metric by an antimetric. Examples are given, showing the complexity of these structures and their prevalence. Among them are: a nonholonomic antimetric on Heisenberg group, an antimetric on a nonnilpotent group admitting dilatations, a pointed locally generated semigroup in the Hilbert space with trivial tangent cone, antinorms connected with the Brunn–Minkowski inequality and Shannon entropy, a discontinuous antinorm on a Lie algebra determining a continuous antimetric on the Lie group, and an example of the converse situation. Problems are formulated. Bibliography: 47 titles.     

An algorithm for routing messages between processing elements in a multiprocessor system which tolerates a maximal number of faulty links

An algorithm for routing messages between processing elements in a multiprocessor system is proposed. The algorithm can always be performed when the system is a connected graph. Comparisons are made with some known algorithms. The algorithm has been illustrated on an n-dimensional hypercube but can be applied on an arbitrary connected multiprocessor system.